$ A = \left[\begin{array}{rrr}-1 & 0 & 1 \\ 3 & 3 & -2\end{array}\right]$ $ B = \left[\begin{array}{rr}5 & 2 \\ 2 & -1 \\ 0 & 4\end{array}\right]$ What is $ A B$ ?
Solution: Because $ A$ has dimensions $(2\times3)$ and $ B$ has dimensions $(3\times2)$ , the answer matrix will have dimensions $(2\times2)$ $ A B = \left[\begin{array}{rrr}{-1} & {0} & {1} \\ {3} & {3} & {-2}\end{array}\right] \left[\begin{array}{rr}{5} & \color{#DF0030}{2} \\ {2} & \color{#DF0030}{-1} \\ {0} & \color{#DF0030}{4}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ A$ , with the corresponding elements in column $j$ of the second matrix, $ B$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ A$ with the first element in ${\text{column }1}$ of $ B$ , then multiply the second element in ${\text{row }1}$ of $ A$ with the second element in ${\text{column }1}$ of $ B$ , and so on. Add the products together. $ \left[\begin{array}{rr}{-1}\cdot{5}+{0}\cdot{2}+{1}\cdot{0} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ A$ with the corresponding elements in ${\text{column }1}$ of $ B$ and add the products together. $ \left[\begin{array}{rr}{-1}\cdot{5}+{0}\cdot{2}+{1}\cdot{0} & ? \\ {3}\cdot{5}+{3}\cdot{2}+{-2}\cdot{0} & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ A$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ B$ and add the products together. $ \left[\begin{array}{rr}{-1}\cdot{5}+{0}\cdot{2}+{1}\cdot{0} & {-1}\cdot\color{#DF0030}{2}+{0}\cdot\color{#DF0030}{-1}+{1}\cdot\color{#DF0030}{4} \\ {3}\cdot{5}+{3}\cdot{2}+{-2}\cdot{0} & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{-1}\cdot{5}+{0}\cdot{2}+{1}\cdot{0} & {-1}\cdot\color{#DF0030}{2}+{0}\cdot\color{#DF0030}{-1}+{1}\cdot\color{#DF0030}{4} \\ {3}\cdot{5}+{3}\cdot{2}+{-2}\cdot{0} & {3}\cdot\color{#DF0030}{2}+{3}\cdot\color{#DF0030}{-1}+{-2}\cdot\color{#DF0030}{4}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}-5 & 2 \\ 21 & -5\end{array}\right] $